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Angle-Side-Angle (ASA) Congruence Theorem

Theorem: Two triangles are congruent if two angles and included side of one triangle are equal to the corresponding two angles and the included side of the other triangle.

Given: In two triangles ABC and PQR, B = Q, C = R and BC = QR

To Prove: ΔABC ΔPQR


Angle-Side-Angle (ASA) Congruence Theorem



Proof:
Case I :
If AB = PQ, ΔABC will be congruent to ΔPQR by the SAS criterion, and the theorem is proved.


Case II : Suppose AB PQ and suppose AB is less than PQ. Take a point S on PQ such that QS = AB. Join RS

In Δ ABC and Δ SQR,
AB = SQ ... (Supposed)
BC = QR ... (Given)

B = Q ... (Given)
ΔABC ΔSQR ... (SAS Criterion)
Hence,
ACB = QRS ... (c.p.c.t.)
But
ACB = QRP ... (Given)
QRP = QRS
which is impossible unless ray RS coincides with ray RP or S coincides with P.
AB must be equal to PQ.


Case III : If we suppose that AB is greater than PQ, a similar argument applies and ΔABC ΔPQR. Hence, in all cases, ΔABC ΔPQR.

 





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